Lecture 10

Estimation with Non-Binary Covariates

Population and Sample

income education sex
1 $51k 14 male
2 $0k 16 female
⋮
14194 $62k 12 male
income education sex
1 $120k 16 male
2 $4k 13 female
⋮
2271 $0k 20 female
  • Today we’re going to talk about estimation when we have non-binary covariates.
  • We’ve got a sample (right) which we’ll think of as being drawn with replacement from a population (left).
  • And we’ll use the sample to estimate various summaries of the population.

Summaries

education income.mean number
8 $17k 86
9 $21k 175
20 $111k 461

\[ \mu(x) = \frac{1}{m_x}\sum_{j:x_j=x} y_j \qfor m_x = \sum_{j:x_j=x} 1 \]

education income.mean number
8 $19k 14
9 $21k 26
20 $108k 71

\[ \hat\mu(x) = \frac{1}{N_{x}}\sum_{i:X_i=x} Y_i \qfor N_{x} = \sum_{i:X_i=x} 1 \]

  • We summarize the relationship between education and income by the mean income for each level of education.
    • When we do this for the population, we get the stuff on the left.
    • When we do this for the sample, we get the stuff on the right.
  • Exercise. What is the mean income for California residents in our population with 10 years of education?
  • Locate it in the plot, write it in mathematical notation, then find the value in the table.

Summaries Based on Multiple Covariates

education sex income.mean number
8 female $3k 40
8 male $29k 46
20 male $114k 204

\[ \mu(w,x) = \frac{1}{m_{w,x}}\sum_{\substack{j\\w_j=w\\x_j=x}} y_j \qfor m_{w,x} = \sum_{\substack{j \\ w_j=w \\ x_j=x}} 1 \]

education sex income.mean number
8 female $4k 5
8 male $28k 9
20 male $110k 35

\[ \hat\mu(w,x) = \frac{1}{N_{w,x}}\sum_{\substack{i \\ W_i=w \\ X_i=x}} Y_i \qfor N_{w,x} = \sum_{\substack{i \\ W_i=w \\ X_i=x}} 1 \]

  • We can work with finer groups defined using multiple covariates. Here, we use sex (\(w\)) and education (\(x\)).
  • Exercise. What is the mean income for female California residents in our population with 9 years of education?
  • Locate it in the plot, write it in mathematical notation, then find the value in the table.

Unbiasedness

sex education income.mean number
female 8 $3k 40
male 8 $29k 46
male 20 $114k 204

\[ \mu(x) = \frac{1}{m_{w,x}}\sum_{\substack{j\\w_j=w\\x_j=x}} y_j \qfor m_{w,x} = \sum_{\substack{j \\ w_j=w \\ x_j=x}} 1 \]

sex education income.mean number
female 8 $4k 5
male 8 $28k 9
male 20 $110k 35

\[ \hat\mu(w,x) = \frac{1}{N_{w,x}}\sum_{\substack{i \\ W_i=w \\ X_i=x}} Y_i \qfor N_{w,x} = \sum_{\substack{i \\ W_i=w \\ X_i=x}} 1 \]

  • Subsample means are, in general, unbiased estimators of the corresponding subpopulation mean.

\[ \mathop{\mathrm{E}}[ \hat \mu(w,x) ] = \mu(w,x) \]

  • The proof is exactly the same as it was for a single binary covariate, so we won’t repeat it here.

Unbiasedness

sex education income.mean number
female 8 $3k 40
male 8 $29k 46
male 20 $114k 204

\[ \mu(x) = \frac{1}{m_{w,x}}\sum_{\substack{j\\w_j=w\\x_j=x}} y_j \qfor m_{w,x} = \sum_{\substack{j \\ w_j=w \\ x_j=x}} 1 \]

sex education income.mean number
female 8 $4k 5
male 8 $28k 9
male 20 $110k 35

\[ \hat\mu(w,x) = \frac{1}{N_{w,x}}\sum_{\substack{i \\ W_i=w \\ X_i=x}} Y_i \qfor N_{w,x} = \sum_{\substack{i \\ W_i=w \\ X_i=x}} 1 \]

  • Linearity of expectations justifies using linear combinations of subsample means to estimate corresponding combinations of subpopulation means.

  • For example, the value of a 4-year degree vs. a high school diploma for male California residents. \[ \mathop{\mathrm{E}}[\hat \mu(\text{m},16)-\hat\mu(\text{m}, 12)] = \mu(\text{m}, 16) - \mu(\text{m}, 12) \]

  • What population summary does this estimate? Describe it in mathematical notation then in words. \[ \qty{\hat\mu(\text{m},16)-\hat\mu(\text{m}, 12)} - \qty{\hat\mu(\text{f},16)-\hat\mu(\text{f}, 12)} \]

Activity

Exercise 1

  • Today we’re going to do a few exercises to practice translating between plain english, plots, mathematical notation, and actual numerical values.
  • They’ll be versions of of the party game `telephone’. The first one is in teams of 4.
  1. Person 1 is given a description, in plain english, of some population summary. They translate it into a description in terms of a plot, write it down, and pass it to person 2.
  2. Person 2 translates this into a mathematical formula, writes it down, and passes it to person 3.
  3. Person 3 translates this back into a description in terms of a plot, writes it down, and passes it to person 4.
  4. Person 4 translates this back into a description in plain english and writes that down.
  • We’ll then compare the original description to the final one to see how well the information was transmitted.
  • If they don’t match, we’ll review the chain of descriptions together to see how well the information was transmitted.

Exercise 2

  • We’ll do a second version of the game with a different end.
  • As before, we’ll start with a description of a population summary in plain english.
  • But this time, we’ll end with an estimate.
  1. Person 1 is given a description, in plain english, of some population summary. They translate it into a description in mathematical notation, write it down, and pass it to person 2.
  2. Person 2 writes down a sample summary that estimates the population one, in mathematical notation, and passes it to person 3.
  3. Person 3 calculates the estimate, writes it down, and passes it to person 4.
  4. Person 4, who was given the same description as Person 1, collects all intermediate descriptions and checks the team’s work.
  • As before, we’ll review these together to see how well the information was transmitted.

Possible Estimation Targets

  • The mean income for California residents with 12-year of education.
  • The difference in mean incomes between male and female California residents with graduate degrees.
  • The average incremental value of 10th, 11th, and 12th grade.
  • The average value, for female California residents with a 10th-12th grade educations, of their last year of education.

Bivariate Sample Summaries

sex education income.mean number
female 8 $4k 5
male 8 $28k 9
female 9 $14k 15
male 9 $30k 11
female 10 $12k 9
male 10 $4k 7
female 11 $11k 40
male 11 $30k 35
female 12 $21k 252
male 12 $35k 353
female 13 $27k 178
male 13 $41k 179
female 14 $30k 110
male 14 $50k 110
female 16 $58k 329
male 16 $70k 311
female 18 $77k 140
male 18 $100k 107
female 20 $107k 36
male 20 $110k 35

Bivariate Population Summaries

sex education income.mean number
female 8 $3k 40
male 8 $29k 46
female 9 $14k 92
male 9 $29k 83
female 10 $13k 70
male 10 $4k 51
female 11 $12k 259
male 11 $30k 230
female 12 $20k 1553
male 12 $34k 2212
female 13 $27k 1064
male 13 $41k 1146
female 14 $32k 689
male 14 $49k 698
female 16 $57k 2021
male 16 $70k 1956
female 18 $77k 874
male 18 $101k 649
female 20 $109k 257
male 20 $114k 204